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%I #43 Oct 09 2020 03:50:59
%S 1,3,7,16,33,63,117,202,344,566,908,1419,2206,3334,4988,7378,10778,
%T 15535,22281,31547,44405,62011,85939,118281,162136,220494,298531,
%U 402163,539181,719301,956287,1265022,1667973,2190934,2867470,3739797,4864163,6303461,8146863,10499087,13493267,17293169,22111954
%N Number of pyramid polycubes of a given volume in dimension 3.
%C A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell of coordinate (0,b,c) belonging to the first plateau is such that b , c >= 0.
%C If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau.
%H C. Carré, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Dubernard/dub4.html">Enumeration of Polycubes and Dirichlet Convolutions</a>, J. Int. Seq. 18 (2015) 15.11.4; also <a href="https://hal.archives-ouvertes.fr/hal-00905889/en">hal-00905889</a>, version 1.
%F The generating function for the numbers of pyramids of height h and volumes v_1 , ... v_h is (n_1-n_2+1) *(n_2-n_3+1) *... *(n_{h-1}-n_h+1) *(x_1^{n_1} * ... x_h^{n_h}) / ((1-x_1^{n_1}) *(1-x_1^{n_1}*x_2^{n_2}) *... *(1-x_1^{n_1}*x_2^{n_2}*...x_h^{n_h})).
%F This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>=, ... >= n_h>=1 and then over h.
%Y A001931 is an upper bound.
%K nonn
%O 1,2
%A _Matthieu Deneufchâtel_, Oct 03 2013
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